SOLUTION: Three cards are dealt from a standard deck of fifty-two cards (without replacement); the probability that there is at least one face card?

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Question 1142552: Three cards are dealt from a standard deck of fifty-two cards (without replacement); the probability that there is at least one face card?
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20065)   (Show Source): You can put this solution on YOUR website!
Three cards are dealt from a standard deck of fifty-two cards (without replacement); the probability that there is at least one face card? 

The entire deck of cards are these 52:


 
A♥   2♥   3♥   4♥   5♥   6♥   7♥   8♥  9♥  10♥  J♥  Q♥  K♥ 
A♦   2♦   3♦   4♦   5♦   6♦   7♦   8♦  9♦  10♦  J♦  Q♦  K♦
A♠   2♠   3♠   4♠   5♠   6♠   7♠   8♠  9♠  10♠  J♠  Q♠  K♠  
A♣   2♣   3♣   4♣   5♣   6♣   7♣   8♣  9♣  10♣  J♣  Q♣  K♣ 

The number of ways to select ANY three cards from those 52 is

52 cards choose 3 = 52C3 = 22100 ways.  That will be the denominator
of our probability before reducing.

But for the complement event, we remove the face cards, which leaves
these cards


 
A♥   2♥   3♥   4♥   5♥   6♥   7♥   8♥  9♥  10♥ 
A♦   2♦   3♦   4♦   5♦   6♦   7♦   8♦  9♦  10♦
A♠   2♠   3♠   4♠   5♠   6♠   7♠   8♠  9♠  10♠  
A♣   2♣   3♣   4♣   5♣   6♣   7♣   8♣  9♣  10♣


So the numer of ways to select three cards from those 40 is

40 cards choose 3 = 40C3 = 9880

So the probability of the complement event is

9880 ways out of 22100 or  which reduces to

  <-- probability of complement event of dealing
NO face cards,

Therefore the probabilkity of selecting at least one face
card is  which equals 

That's close to 0.55

Edwin
Answer by greenestamps(13215)   (Show Source): You can put this solution on YOUR website!


P(at least one face card) = 1 - P(no face cards)

So calculate the probability of getting no face cards when 3 cards are drawn:

P(first card not a face card) = 40/52
P(second card not a face card) = 39/51
P(third card not a face card) = 38/50

P(no face cards) = (40/52)(39/51)(38/50) = .447 to 3 decimal places

So P(at least one face card = 1-.447 = .553
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